Though the diet also includes skim or low-fat dairy products, the American Academy of Pediatrics recommends that children consume whole milk until age 2, Johnson said.

Nuts are also included in the Mediterranean diet, but the academy considers whole nuts to be a high-risk food for choking and recommends caution for children under age 4, Johnson said. And, though the Mediterranean diet includes a modest amount of wine, alcohol is illegal to consume until age 21 in the United States.

With those caveats, however, how exactly might a Mediterranean diet benefit children's health? And how can parents get their kids eating Mediterranean? Here are some of the latest findings and tips from experts.

How a Mediterranean diet can be kid-healthy

In adults, the Mediterranean diet has been linked to improved brain, heart and bone health; longer life; and a lower risk of breast cancer and type 2 diabetes.

For children, some studies suggest that the diet might be associated with a reduced risk for obesity, asthma and allergies; there might also be an association between not eating the diet and attention deficit hyperactivity disorder.

A Mediterranean-style diet was linked to a significant decrease in body mass index, fat mass and glucose levels in children with obesity in a small study in the journal BMC Pediatrics in 2014.

The study, which was conducted in Mexico, compared the health of 24 obese children who were assigned to eat a Mediterranean-style diet with 25 who were assigned to eat a standard diet over a 16-week period.

"These results further support the importance of introducing a Mediterranean-style diet to at-risk populations," the researchers wrote.

Another study, presented at the European Congress on Obesity that year, found that children who consumed a Mediterranean diet were 15% less likely to be overweight or obese than those children who did not.

Other studies have explored how a Mediterranean diet might be associated with reduced asthma and allergy symptoms in children. Adherence to a Mediterranean diet tended to be associated with a lower occurrence of wheezing and asthma in a 2013 paper in the journal Pediatric Allergy and Immunology.

For that paper, researchers analyzed seven epidemiological studies on Mediterranean-like diets and asthma symptoms that were published between 1960 and 2012.

"The results of the present meta-analysis show that adherence to a Mediterranean diet during childhood is a protective factor for 'current wheeze' and 'asthma ever,' but not for 'severe current wheeze' when considering all studies together," the researchers wrote in the paper.

Additionally, a study of 695 children found that those whose mothers took fish oil supplements in the third trimester of pregnancy had a reduced risk -- by about 7 percentage points -- for wheezing, asthma or lower respiratory tract infections. The study was published last year in The New England Journal of Medicine.

Fatty fish is a regular part of the Mediterranean diet.

A small study published in the journal Pediatrics on Monday suggests that eating less of a Mediterranean diet -- meaning fewer vegetables, fruits and fish, but more junk food -- might be associated with a 7.07-fold higher risk of ADHD diagnoses in children.

"The mechanisms linking low-quality diet and ADHD are still unknown, but an unbalanced diet can lead to deficiencies in essential nutrients, such as iron, zinc, magnesium or omega-3 fatty acids, which contribute to healthy cognitive and physical growth," said Maria Izquierdo-Pulido, a food science and nutrition researcher at the University of Barcelona in Spain and a co-author of the study.

Those nutrients also seem to play an essential role in the development of ADHD, she said.

Yet other experts remain dubious that the new study provides support for the idea that a Mediterranean diet might play a role in ADHD development.

"I have no special tips for parents who want their kid with ADHD to go on the Mediterranean diet, because there is no reason to believe that such a diet is going to be any more helpful for alleviating their ADHD symptoms than any basically sound diet," said Thomas Brown, adjunct clinical associate professor at the University of Southern California's Keck School of Medicine and director of the Brown Clinic for Attention and Related Disorders in Hamden, Connecticut. He was not involved in the new study.

"The hypothesis that those with ADHD would be less adherent to the Mediterranean diet is puzzling because it is well-known that, as a group, those with ADHD tend to be less adherent to many tasks and routines they are asked to do," Brown said. "ADHD is essentially an inherited problem with the brain's self-management system."

How to get your child to eat 'Mediterranean'

For parents who would like to introduce a Mediterranean diet into their children's daily eating routine, Dr. Carolina Vidal, an assistant professor of psychiatry and behavioral sciences at Johns Hopkins Medicine, offered a few tips.

"Slowly introduce fruits and vegetables, and present them consistently with the other foods they eat. Cutting out sugars and fast food meals and make those sporadic as opposed to a regular meal or snack is very helpful," she said. "Sitting to eat with them and serving as a role model is the best way to introduce a Mediterranean diet."

"Exposure to foods leads to a preference for those foods," she added. "It can take up to 12 to 15 exposures to a new food for a child to develop a preference for that food. Ask your child to taste the food, and don't give up too soon."